Optimal. Leaf size=107 \[ -\frac {\left (4 a c+3 b^2\right ) \tan ^{-1}\left (\frac {b-2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{16 c^{5/2}}-\frac {3 b \sqrt {a+b x^2-c x^4}}{8 c^2}-\frac {x^2 \sqrt {a+b x^2-c x^4}}{4 c} \]
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Rubi [A] time = 0.09, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1114, 742, 640, 621, 204} \[ -\frac {\left (4 a c+3 b^2\right ) \tan ^{-1}\left (\frac {b-2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{16 c^{5/2}}-\frac {3 b \sqrt {a+b x^2-c x^4}}{8 c^2}-\frac {x^2 \sqrt {a+b x^2-c x^4}}{4 c} \]
Antiderivative was successfully verified.
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Rule 204
Rule 621
Rule 640
Rule 742
Rule 1114
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {a+b x^2-c x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b x-c x^2}} \, dx,x,x^2\right )\\ &=-\frac {x^2 \sqrt {a+b x^2-c x^4}}{4 c}-\frac {\operatorname {Subst}\left (\int \frac {-a-\frac {3 b x}{2}}{\sqrt {a+b x-c x^2}} \, dx,x,x^2\right )}{4 c}\\ &=-\frac {3 b \sqrt {a+b x^2-c x^4}}{8 c^2}-\frac {x^2 \sqrt {a+b x^2-c x^4}}{4 c}+\frac {\left (3 b^2+4 a c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x-c x^2}} \, dx,x,x^2\right )}{16 c^2}\\ &=-\frac {3 b \sqrt {a+b x^2-c x^4}}{8 c^2}-\frac {x^2 \sqrt {a+b x^2-c x^4}}{4 c}+\frac {\left (3 b^2+4 a c\right ) \operatorname {Subst}\left (\int \frac {1}{-4 c-x^2} \, dx,x,\frac {b-2 c x^2}{\sqrt {a+b x^2-c x^4}}\right )}{8 c^2}\\ &=-\frac {3 b \sqrt {a+b x^2-c x^4}}{8 c^2}-\frac {x^2 \sqrt {a+b x^2-c x^4}}{4 c}-\frac {\left (3 b^2+4 a c\right ) \tan ^{-1}\left (\frac {b-2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{16 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 89, normalized size = 0.83 \[ -\frac {\left (4 a c+3 b^2\right ) \tan ^{-1}\left (\frac {b-2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{16 c^{5/2}}-\frac {\left (3 b+2 c x^2\right ) \sqrt {a+b x^2-c x^4}}{8 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 211, normalized size = 1.97 \[ \left [-\frac {{\left (3 \, b^{2} + 4 \, a c\right )} \sqrt {-c} \log \left (8 \, c^{2} x^{4} - 8 \, b c x^{2} + b^{2} - 4 \, \sqrt {-c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} - b\right )} \sqrt {-c} - 4 \, a c\right ) + 4 \, \sqrt {-c x^{4} + b x^{2} + a} {\left (2 \, c^{2} x^{2} + 3 \, b c\right )}}{32 \, c^{3}}, -\frac {{\left (3 \, b^{2} + 4 \, a c\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} - b\right )} \sqrt {c}}{2 \, {\left (c^{2} x^{4} - b c x^{2} - a c\right )}}\right ) + 2 \, \sqrt {-c x^{4} + b x^{2} + a} {\left (2 \, c^{2} x^{2} + 3 \, b c\right )}}{16 \, c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 91, normalized size = 0.85 \[ -\frac {1}{8} \, \sqrt {-c x^{4} + b x^{2} + a} {\left (\frac {2 \, x^{2}}{c} + \frac {3 \, b}{c^{2}}\right )} - \frac {{\left (3 \, b^{2} + 4 \, a c\right )} \log \left ({\left | 2 \, {\left (\sqrt {-c} x^{2} - \sqrt {-c x^{4} + b x^{2} + a}\right )} \sqrt {-c} + b \right |}\right )}{16 \, \sqrt {-c} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 120, normalized size = 1.12 \[ -\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, x^{2}}{4 c}+\frac {a \arctan \left (\frac {\left (x^{2}-\frac {b}{2 c}\right ) \sqrt {c}}{\sqrt {-c \,x^{4}+b \,x^{2}+a}}\right )}{4 c^{\frac {3}{2}}}+\frac {3 b^{2} \arctan \left (\frac {\left (x^{2}-\frac {b}{2 c}\right ) \sqrt {c}}{\sqrt {-c \,x^{4}+b \,x^{2}+a}}\right )}{16 c^{\frac {5}{2}}}-\frac {3 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, b}{8 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.42, size = 105, normalized size = 0.98 \[ -\frac {\sqrt {-c x^{4} + b x^{2} + a} x^{2}}{4 \, c} - \frac {3 \, b^{2} \arcsin \left (-\frac {2 \, c x^{2} - b}{\sqrt {b^{2} + 4 \, a c}}\right )}{16 \, c^{\frac {5}{2}}} - \frac {a \arcsin \left (-\frac {2 \, c x^{2} - b}{\sqrt {b^{2} + 4 \, a c}}\right )}{4 \, c^{\frac {3}{2}}} - \frac {3 \, \sqrt {-c x^{4} + b x^{2} + a} b}{8 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^5}{\sqrt {-c\,x^4+b\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\sqrt {a + b x^{2} - c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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